Ill-posedness of a double null free evolution scheme for black hole spacetimes

TL;DR

This paper demonstrates that free evolution schemes for Einstein's equations, which do not enforce constraints, can develop exponential instabilities, making them unreliable for simulating black hole spacetimes.

Contribution

It shows that unconstrained free evolution methods may be fundamentally ill-posed due to exponentially growing modes in black hole simulations.

Findings

Free evolution schemes can contain exponentially growing modes.

Such schemes may be inherently unstable for black hole spacetimes.

Constraint enforcement might be necessary for stable numerical evolution.

Abstract

We suggest that ``free evolution'' integration schemes for the Einstein equations (that do not enforce constraints) may contain exponentially growing modes that render them useless in numerical integrations of black hole spacetimes, independently of how the equations are differenced. As an example we consider the evolution of Schwarzschild and Reissner-Nordstr\"om spacetimes in double null coordinates.